The number of honeybees in a beehive as a function of the number of weeks that have passed can be modelled by the logistic function
B(w) = 3000/1+1499(0.4)^w
What is the maximum number of honeybees that can exist in the beehive?
as w gets huge, 0.4^w gets small, and approaches zero
so the maximum number of bees is 3000/(1+0) = 3000
To find the maximum number of honeybees that can exist in the beehive, we need to determine the asymptote of the logistic function.
In the given function B(w) = 3000/(1 + 1499(0.4)^w), the denominator (1 + 1499(0.4)^w) approaches 1 as w increases to infinity. This means that as w becomes very large, the value inside the parentheses becomes negligible compared to 1.
Therefore, as w approaches infinity, the logistic function approaches B(w) = 3000/1 = 3000.
Hence, the maximum number of honeybees that can exist in the beehive is 3000.
To find the maximum number of honeybees that can exist in the beehive, we need to determine the limiting value of the logistic function as the number of weeks approaches infinity.
In the given logistic function B(w) = 3000/(1 + 1499*(0.4)^w), the coefficient of the term (0.4)^w is less than 1, indicating that the function approaches a limiting value as w goes to infinity. The limiting value is the maximum number of honeybees that can exist in the beehive.
To calculate the limiting value, we substitute infinity for the variable w in the function:
lim(w→∞) B(w) = lim(w→∞) [3000/(1 + 1499*(0.4)^w)]
Since (0.4)^w approaches zero as w goes to infinity, the limiting value of the function can be calculated as:
lim(w→∞) B(w) = 3000/1 = 3000
Therefore, the maximum number of honeybees that can exist in the beehive is 3000.